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Harmonic number

The th harmonic number is defined as the truncation of the harmonic series after the th term

The first ten harmonic numbers are:

.

To compute more harmonic numbers visit .

is integral only for [1] .

To see the first statement note that the greatest power of 2 not exceeding do not divide any other term if . Therefore the numerator of must be odd while the denominator is even. In other words, the denominator of is a multiple of where is determined by relations .

L.Kürschák [2] proved more: the difference between two distinct harmonic numbers is never an integer. He proved this in the form that with is never an integer.

T.Nagell [3] proved that even , , cannot be an integer. This result was later reproved by P.Erdõs [4] .

The Bertrand postulate says that for every positive integer there is a prime such that . Consequently, if the denominator of is divisible by at least one odd prime. As we saw above, the denominator of is divisible by the largest power of 2 less or equal to . Consequently, is the only harmonic number with a prime denominator.

An analogous question for which the numerator of is a prime number is more delicate. The first 10 values of are (for more details visit ):

where denotes the unsigned Stirling number of the first kind, counting the permutation of elements that are product of disjoint cycles.

Euler found the following integral representation

(3)

It can be shown using the integral comparison criterion applied to the function for that

(4)

This implies that , and that , that is that grows about as fast (or better as slowly) as . Actually more can be shown. Using the fact that converges in the infinite product there follows that there exists a non-vanishing finite limit

(5)

say . Clearly . Consequently there exists also the limit

(6)

The constant is the Euler-Mascheroni constant . To see this we can use Euler’s summation formula which gives that

(7)

where stands for the th Bernoulli number. This formula can be used in several ways:

to get an asymptotic approximations for , for instance for we get

For instance if we get that without adding up 1000 unit fractions.

to get an asymptotic approximation for . Previous formula with yields (cf. also )

to discover and prove various asymptotic relations, for instance that for every we have

Last statement can be extended also in the following form: is monotone increasing and converges to. Since

we have

Moreover

using the substitution .

Graphical representation of values (red points) of and their asymptotic curve (blue line)

A generating function for harmonic numbers is

(8)

If denotes the number of divisors of an integer then

In the following graph the harmonic numbers are represented by red points and the values of right hand side by blue points

J.C.Lagarias [6] proved an interesting link to the Riemann hypothesis. He proved that the problem of showing that

(9)

for all with equality only for , is equivalent to the Riemann Hypothesis. The proof uses two bounds for the sums-of-divisors function proved by G. Robin [7] . One of the inequalities says that the inequality for is equivalent with Riemann hypotheses (Robin also showed that one can take ).

In the graph the blue points represent the sum-of-divisors functions and the red ones the values of the right hand side of (9)

For the first multiples of 12 we have

It is possible to extend the above definition of a harmonic number also to non-integral values of using the observation that

(10)

converges for all complex values of , except when is a negative number. If denotes the value of the series, then when is a positive integer we get .

Wolstenholme's Theorem: If is a prime, then [8]

the numerator of the harmonic number is divisible by , that is

the numerator of the generalized harmonic number is divisible by , that is

An interesting problem in which solution harmonic numbers unexpectedly occur was discovered by R.T.Sharp [9][10] : determine how far an overhang we can achieve by stacking dominoes over the table edge, accounting for the force of gravity.

The center of gravity is a geometric property of any object and it is the point where all the weight of the object can be considered to be concentrated. In other words, it is defined as the average location of the weight of an object. Thus the radiusvector of the center of a system of particles is the average of the positions of center of the th particle weighted by its mass

(11)

Since we shall suppose that edges of the cards are parallel to the edge of the table, we shall only work with the -coordinate. Let be the shift distance of the extreme edge of the th domino to the corresponding edge of the st domino piece counting from the top.

For the sake of simplicity suppose that length of each domino piece is 2 units, the weight is 1 unit with uniformly distributed mass along the length,, and that the pile contains pieces. Then the -coordinate of the center of gravity of th piece from the top is , . Then the -coordinate of the center of such complex of superimposed domino pieces is

(12)

while .

The stability criterion is that the -coordinate of the center of gravity of pieces must lie exactly above the right edge of piece . Since the right edge of piece has -coordinate 2, we get from (12)

(13)

This implies that for . This implies that the total extension for block of dominoes is . In the following picture the th triangle shaped point from top always shows the position of the center of gravity of top dominoes.

A bit surprising consequence of the solution is that given sufficiently large number of dominoes, arbitrary large overhangs can be achieved.

In the above way of stacking of dominoes we achieved an overhang of , We stacked here all the dominoes one on top of another with the th domino from the top displaced by beyond the block below. This solution is however not optimal. M.Paterson and U.Zwick [11] showed that it is exponentially far from optimal and gave explicit constructions with an overhang of .

Generalized harmonic number

Generalized harmonic numberof order is defined as truncation of the series of reciprocals of the th powers of positive integers

The generalized harmonic number converges to the Riemann zeta function in the following sense

Hyperharmonic number

J.H.Conway and R.K.Guy [12] defined the hyperharmonic numbers, the th harmonic number of order: , and for .

The first ten harmonic numbers of order 2 are:

The first ten harmonic numbers of order 3 are:

We have [13]

(15)

where is the number of permutations of the set having disjoint non-empty cycles in which the elements are restricted to appear in different cycles (the so-called -Stirling numbers [14] , where ), or [12]